Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.

We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, A n q,m . Included in this class are the truncated polynomial algebras k[X 1 , . . . , X m ]/(X n i ), with k an algebraically closed field and char(k) arbitrary. We prove that these varieties characterise projectivity of modules (Dade's lemma) and examine the implications for the tree class of the stable Auslander-Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade's lemma in this context.

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We consider algebras K[X 1 , . . . , X m ]/(X 2 i ), where K is an algebraically closed fields. To any finite dimensional module for this algebra we associate a rank variety. When char(K) = 2 we recover Carlson's rank variety. The main result states that a module is projective if and only if its rank variety vanishes. This has applications to other algebras, including tensor products of certain Brauer tree algebras and certain parabolic Hecke algebras. In addition, the result has implications for the graph structure of the stable Auslander-Reiten quiver.